I am having trouble finishing the derivation using the Milne-Thompson Circle Theorem (MTCT) for a potential flow with a vortex at a location $\zeta_v$.
My understanding of the MTCT is that the complex potential $w_v$ (with subscript $v$ for vortex) transforms accordingly when a circle is present,
$$w_{vc}(\zeta)=w_v(\zeta)+\overline{w_v\left(\frac{a^2}{\overline{\zeta}}\right)}$$ or equivalently $$w_{vc}(\zeta)=w_v(\zeta)+\overline{w_v}\left(\frac{a^2}{\zeta}\right).$$
My complex velocity is $$w_v(\zeta)=\frac{\Gamma}{2\pi i}\ln(\zeta-\zeta_v)$$ where $\zeta_v$ is the position of the vortex. I know the final complex velocity with the circle $w_{vc}$ is $$w_{vc}(\zeta)=\frac{\Gamma}{2\pi i}\left[\ln(\zeta-\zeta_v)-\ln\left(\zeta-\frac{a^2}{\overline{\zeta_v}}\right)+\ln(\zeta)\right].$$ The furthest I've been able to get is the following, $$\frac{\Gamma}{2\pi i}\left[\ln(\zeta-\zeta_v)-\ln\left(a^2\left(\frac{1}{\zeta}-\frac{1}{\zeta_v}\right)\right)\right]$$ and I'm not sure if I made a mistake or if I just can't figure what to do with this form of the equation.
I think some of my confusion may be because of notation. My textbook by Batchelor says the MTCT uses $\overline{w\left(\frac{a^2}{\zeta}\right)}$ where elsewhere it is $\overline{w\left(\frac{a^2}{\overline{\zeta}}\right)}=\overline{w}\left(\frac{a^2}{\zeta}\right)$ which I guess are identical. Although, the answers to these two questions (1, 2) seem to contradict that but I'm probably misunderstanding something.
Thank you very much.
Best Answer
I can't reconcile the last term, but here's what I have so far:
$w_{vc}(\zeta) = w_v(\zeta)+\bar{w}_v(\frac{a^2}{\zeta})$
$=\frac{\Gamma}{2\pi i}[ln(\zeta-\zeta_v)-ln(\frac{a^2}{\zeta}-\bar{\zeta}_v)]$
Since:
$ln(\frac{a^2}{\zeta}-\bar{\zeta}_v)=ln(\zeta-\frac{a^2}{\bar{\zeta}_v})+ln(-\frac{\bar{\zeta}_v}{\zeta})=ln(\zeta-\frac{a^2}{\bar{\zeta}_v})+ln(-\bar{\zeta}_v)-ln(\zeta)$
We have that:
$w_{vc}(\zeta) = \frac{\Gamma}{2\pi i}[ln(\zeta-\zeta_v)-ln(\zeta-\frac{a^2}{\bar{\zeta}_v})+ln(\zeta)-ln(i\bar{\zeta}_v)-\frac{i\pi}{2}]$
Maybe the last terms go away because it's constant, and these are periodic functions? I also could have messed up somewhere; I'm not the best with complex logarithms.